A natural requirement in relativistic theories is that the Lagrangian density must be a Lorentz scalar. This assures the validity of the Poincaré group, which is the fundamental transformation group on Minkowski space. In another context, in gauge theories, interactions are obtained from local gauge invariance principles. Usually, the local gauge symmetry is postulated in advance. Afterward, the action invariance is explicitly verified [1]}, [2]}. The question, regarding an opposite point of view, is how to systematically derive Noether symmetries uniquely from the Lagrangian density, without additional hypothesis. The advantage of the deductive approach is the possibility of unveiling symmetries not apparent from the very beginning. In addition, given a broad class of Lagrangians, one might be interested in the identification of the particular subclasses admitting Noether symmetries. In Section 7, this approach is put forward in the case of the charged scalar particle under prescribed external electromagnetic fields. In this case, not necessarily the external field will be compatible with some symmetry transformation. It should be mentioned that the traditional approach, put forward e.g. in authoritative textbooks
[1]}, [2]}, [5]}, [6]} is of course perfectly well justified, as long as the overall symmetry structure of space-time and internal space is concerned. However, sometimes (as in the aforementioned case of the charged scalar particle under external fields) one might be interested in subclasses admitting Noether point symmetry, as a toll in the search for sufficiently simple benchmark systems.
